Extension of the matrix Bartlett's formula to the third and fourth order and to noisy linear models with application to parameter estimation

• Published in: IEEE transactions on signal processing Vol. 53; no. 8; pp. 2765 - 2776
• Main Authors: Delmas, J.-P., Meurisse, Y.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.08.2005; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Additive noise; Applied sciences; Asymptotic properties; Bartlett's formula; Calculus; Closed-form solution; Covariance; Covariance matrix; Detection, estimation, filtering, equalization, prediction; Engineering Sciences; Exact sciences and technology; Exact solutions; fourth-order cumulant; Gaussian noise; High level languages; Information, signal and communications theory; Mathematical analysis; Mathematical models; noisy linear process; Parameter estimation; Performance analysis; Representations; Signal and communications theory; Signal and Image processing; Signal processing algorithms; Signal to noise ratio; Signal, noise; statistical performance analysis; Telecommunications and information theory; third-order cumulant; Additive noise; Performance evaluation; ARMA model; Parameter estimation; Bartlett's formula; Third order; Asymptotic behavior; statistical performance analysis; Color noise; noisy linear process; Algorithm; Linear model; fourth-order cumulant; Statistical method; Linear process; third-order cumu lant; Covariance; High level language; S matrix; Matrix method; Cumulant; Signal to noise ratio; Statistical performance analysis; Third-order cumulant; Fourth-order cumulant; Noisy linear process
• ISSN: 1053-587X; 1941-0476; 1941-0476
• DOI: 10.1109/TSP.2005.850362

This paper focuses on the extension of the asymptotic covariance of the sample covariance (denoted Bartlett's formula) of linear processes to thirdand fourth-order sample cumulant and to noisy linear processes. Closed-form expressions of the asymptotic covariance and cross-covariance of the sample second-, third-, and fourth-order cumulants are derived in a relatively straightforward manner, thanks to a matrix polyspectral representation and a symbolic calculus akin to a high-level language. As an application of these extended formulae, we underscore the sensitivity of the asymptotic performance of estimated ARMA parameters by an arbitrary third- or fourth-order-based algorithm with respect to the signal-to-noise ratio, the spectra of the linear process, and the colored additive noise.